Difference between Euclidean space and vector space?

I often hear them used interchangeably ... they are very complicated to make any use of.

Wikipedia words:

Euclidean space:

One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance and angle.

Vector space:

A vector space is a mathematical structure formed by a collection of elements called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars in this context.

They are not related at all. A vector space is a structure composed of vectors and has no magnitude or dimension, whereas Euclidean space can be of any dimension and is based on coordinates.

I hear 3-D programming uses vectors, so Euclidean geometry should be useless, no?

Basically, aren't they unrelated?

A concise mathematical term to describe the relationship between the Euclidean space $X = \mathbb E^n$ and the real vector space $V = \mathbb R^n$ is to say that $X$ is a principal homogeneous space (or ''torsor'') for $V$. This is a way of saying that they are definitely not the same objects, but they very much are related to each other.

In particular:

• The objects $\mathbb E^n$ and $\mathbb R^n$ are exactly the same as sets of elements -- they both correspond bijectively to $n$-tuples of real numbers.

However,

• in the vector space $\mathbb R^n$ we are allowed to add any two vectors (using the ''tip to tail'' visualization), whereas in Euclidean space $\mathbb E^n$ there is no natural way to describe the process of ''adding'' two points. Instead, given two points $P,Q$ in $\mathbb E^n$ we can naturally define their difference $\vec{v} = P-Q$, which is a vector in $ \mathbb R^n$. This vector tells us how to get from point $Q$ to $P$ in $\mathbb E^n$.

• in $\mathbb R^n$ there is a special ''zero vector'' $\vec{0} = (0,\ldots,0)$ which satisfies the additive property $\vec{0} + \vec{v} = \vec{v}$ for any $\vec{v}\in \mathbb R^n$, while in $\mathbb E^n$ there is no point that is somehow more special that the other ones -- i.e. the space is ''homogeneous'' meaning it looks the same around every point.

• for a vector $\vec{v}\in \mathbb R^n$ we can compute its length (or ''magnitute'' or ''norm'' etc.) by the formula $$|\vec{v}| = \sqrt{v_1^2 + \cdots + v_n^2}.$$ For a point $P\in \mathbb E^n$, it does not make sense to ask what its length or distance is. It only makes sense to ask the distance between two points $P,Q$.

There are many natural examples of torsors motivated by physics, discussed in this blog post of John Baez, as well as more rigorous definitions if you are interested.

While a vector space is something very formal and axiomatic, Euclidean space has not a unified meaning. Usually, it refers to something where you have points, lines, can measure angles and distances and the Euclidean axioms are satisfied. Sometimes it is identified with $\mathbb{R}^2$ resp. $\mathbb{R}^n$ but more as an affine and metric space (you have both points and vectors, not just vectors). So, the Euclidean space has softer meaning and usually refers to a richer structure.